Let \((u_n,v_n)\)  be the unique solution to (REF ) given by Lemma REF . By Lemma REF  we have that, up to a subsequence, the following convergences hold as \(n\)  goes to \(+\infty \) :
\(\left\lbrace \begin{array}{lll}\displaystyle v_{n,i} \rightharpoonup v_i & \text{weakly in } L^2(I;H^1(0,1)), \; i = 1,2, \\[2mm]\displaystyle v_{n,0} \rightharpoonup v_0 & \text{in the } L^\infty (I;H^1(0,1))\text{-weak-}\star \; \text{sense},\\[2mm]\displaystyle U_n \rightharpoonup u & \text{weakly in } L^2(I;H^*)\text{ and } H^1(I;V^*),\\[2mm]\displaystyle U_n(t) \rightharpoonup u(t) & \text{weakly in } {V^*}, \text{ for every } t\in I,\\[2mm]\displaystyle \Delta _n u_n \rightharpoonup \dot{u} & \text{weakly in } L^2(I;V^*).\end{array}\right.\) 
Note that, since \(U_n(0)=u^\text{in}\) , we have \(u(0)=u^\text{in}\) . Moreover, by definition of \(U_n\)  we have
\(\Vert U_n-u_n\Vert _{L^2(I;V^*)}\le k_n\Vert \Delta _n u_n\Vert _{L^2(I;V^*)} \rightarrow 0,\ \text{ as }n\rightarrow +\infty .\) 
Hence, for almost every \(t\)  in \(I\) ,
\((U_n-u_n)(t) \underset{n\rightarrow +\infty }{\longrightarrow }0 \text{ in } V^*\\[2mm]\qquad \text{and} \qquad u_n(t) \ \underset{n\rightarrow +\infty }{-\hspace{-4.0pt}-\hspace{-4.0pt}\rightharpoonup } u(t)\; \text{weakly in } {V^*.}\) 
By construction we have that
\(u_{n,i}={\overline{u}}_ie_i(T_M v_{n,i}), \ i=1,2.\) 
Then since \(e_i \circ T_M\)  are Lipschitz continuous, the bounds on \(v_{n,i}\)  ensure that \(u_{n,i}\)  are bounded in \(L^2(I;H^1(0,1))\) .
We deduce then from the nonlinear Aubin-Simon compactness result [1]} that
\(u_{n,i}\rightarrow u_i\ \text{ almost everywhere in }I\times (0,1), \text{ with } u_i={\overline{u}}_ie_i(T_M v_i), \ i=1,2,\) 
which, thanks to (REF ), gives also \(u_{n,0}\rightarrow u_0\)  almost everywhere in \(I\times (0,1)\)  with
\(u_0 = \sum _{i=1,2} z_i u_i + \rho _\text{\rm hl}.\) 
Together with (REF ) and (REF ), this implies that
\(u_{n}\rightarrow u\ \text{ strongly in }L^2(I;L^2(0,1))\text{ and }L^2(I;V^*),\) 
due to the Lebesgue dominated convergence theorem, as well as in the \(L^\infty ((0,1) \times (0,T))\) -weak-\(\star \)  sense.
Moreover, in view of (REF ), one has
\(U_{n,i}\underset{n\rightarrow +\infty }{\longrightarrow }u_i \text{ strongly in }L^2(I;(H^1(0,1))^*),\ i=1,2.\) 
Since \(u_{n,0} = {(E_M v_n)}_0\) , one has
\(\lambda ^2\int _0^1 \partial _x v_{n,0} \partial _x \tilde{v}dx + \sum _{\Gamma \in \lbrace 0,1\rbrace }\left[\left(\beta ^\Gamma v_{n,0}-f^\Gamma \right)\tilde{v}\right](\Gamma )=\int _0^1 u_{n,0} \tilde{v}dx, \qquad \forall \, \tilde{v} \in V.\) 
The aforementioned convergence properties are sufficient to pass to the limit \(n\rightarrow +\infty \)  in the above equality, leading to
\(\lambda ^2\int _0^1 \partial _x v_{0} \partial _x \tilde{v}dx + \sum _{\Gamma \in \lbrace 0,1\rbrace }\left[\left(\beta ^\Gamma v_{0}-f^\Gamma \right)\tilde{v}\right](\Gamma )=\int _0^1 u_{0} \tilde{v}dx, \qquad \forall \, \tilde{v} \in V,\) 
or equivalently \(u_0 = {(E_Mv)}_0\) .
Choosing \(\tilde{v}= v_{n,0}\)  as a test function in (REF ) and passing to the limit \(n\rightarrow +\infty \)  provides
\(\lambda ^2\int _0^T\int _0^1 \left| \partial _x v_{n,0} \right|^2 dxdt &+\; \int _0^T \sum _{\Gamma \in \lbrace 0,1\rbrace } \beta ^\Gamma \left| v_{n,0} \right|^2 dt \\ &= \int _0^T \int _0^1 u_{n,0} v_{n,0}dxdt + \int _0^T \sum _{\Gamma \in \lbrace 0,1\rbrace } f^\Gamma v_{n,0} dt\\&\underset{n\rightarrow +\infty }{\longrightarrow }\int _0^T \int _0^1 u_{0} v_{0}dx dt + \int _0^T\sum _{\Gamma \in \lbrace 0,1\rbrace } f^\Gamma v_{0} dt\\&= \lambda ^2\int _0^T\int _0^1 \left| \partial _x v_{0} \right|^2 dx dt + \int _0^T \sum _{\Gamma \in \lbrace 0,1\rbrace } \beta ^\Gamma \left| v_{0} \right|^2 dt\) 

thanks to (REF ). As a consequence, \(\Vert v_{n,0}\Vert _{L^2(I;H^1(0,1))}\)  tends to \(\Vert v_{0}\Vert _{L^2(I;H^1(0,1))}\) , whence
\(v_{n,0} \underset{n\rightarrow +\infty }{\longrightarrow }v_0 \quad \text{strongly in} \; L^2(I;H^1(0,1)).\) 
Next, due to (REF ) and to (REF ) combined with (REF ) and (REF ),
one gets that
\(\Psi (u_n) \underset{n\rightarrow +\infty }{\longrightarrow }\Psi (u) \quad \text{in}\; L^1(I).\) 
Due to (REF ), we infer that (REF ) holds true.
Finally we want to show that \((u,v)\)  is a solution to problem (REF ).
First we prove that
\(\lim _{n\rightarrow +\infty } A_M(\tau _n v_n,v_n)=A_M(v,v) \text{ in }L^2(I; V^*).\) 
Taking a test function \(\tilde{v} \in V\)  and setting, for \(i=1,2\) , \(\xi _{n,i}=v_{n,i}+z_i v_{n,0}\) , \(\tilde{\xi }_i=\tilde{v}_i +z_i \tilde{v}_0\) , we have
\(\displaystyle \left< A_M(\tau _n v_n,v_n),\tilde{v}\right>=\sum _{i=1,2}\int _0^1 \sigma _i(T_M \tau _n v_{n,i}) \partial _x \xi _{n,i} \partial _x \tilde{\xi }_i dx\\\displaystyle +\sum _{i=1,2}\sum _{\Gamma \in \lbrace 0,1\rbrace }\left[ r_i^\Gamma (T_M \tau _n v_{n,i})g_i^{\Gamma ,\mu }(\xi _{n,i}-\xi _i^{\Gamma })\tilde{\xi }_i\right](\Gamma ).\) 
By (REF ), (REF ) and the properties of the translation operator function \(\tau _n\)  (see for instance [2]})
we have
\(T_M\tau _n v_{n,i} \rightarrow T_M v_i \hbox{ strongly in } L^2(S;L^2(0,1)) \hbox{ and in } L^2_\text{loc}(S;H^s(\Gamma )),\ i=1,2, s>\frac{1}{2},\) 
implying that
\(\sigma _i(T_M \tau _n v_{n,i})\rightarrow \sigma _i(T_M v_{i})\hbox{ strongly in } L^2(S;L^2(0,1)),\ i=1,2,\) 
since \(\sigma _i \circ T_M\)  is Lipschitz continuous,
and
\(r_i^\Gamma (T_M \tau _n v_{n,i})\rightarrow r_i^\Gamma (T_M v_{i})\hbox{ strongly in } L^2(S),\ i=1,2,\ \Gamma \in \lbrace 0,1\rbrace \) 
thanks to the trace theorem.
Moreover, by (REF ) we obtain
\(\xi _{n,i} \rightharpoonup \xi _i \hbox{ weakly in } L^2(S;V)\) 
with \(\xi _i=v_i+z_i v_0\) .
This, together with (REF ), gives
\(\lim _{n\rightarrow +\infty }\sum _{i=1,2}\int _0^1 \sigma _i(T_M \tau _n v_{n,i}) \partial _x \xi _{n,i} \partial _x \tilde{\xi }_i dx=\sum _{i=1,2}\int _0^1 \sigma _i(T_M v_{i}) \partial _x \xi _{i} \partial _x \tilde{\xi }_i dx,\) 
We are now interested in the weak convergence of the term \(g_i^{\Gamma ,\mu }(\xi _{n,i}-\xi _i^{\Gamma })\) .
The approximation \(g_i^{\Gamma ,\mu }\)  of \(g_i^\Gamma \)  has been tailored in (REF ) so that the function \(g_i^{\Gamma ,\mu , NL}\)  defined by
\(g_i^{\Gamma ,\mu , NL}(\xi ) = g_i^{\Gamma ,\mu }(\xi ) - \left(g_i^{\Gamma ,\mu }\right)^{\prime }(\mu ) \, \xi \) 
is constant outside \([-\mu ,\mu ]\) .
In view of the uniform estimate (REF ) of \(v_{n,0}\)  and the one-dimensional Sobolev-Morrey inequality (recall the definition (REF ) of the \(H^1(0,1)\) -norm),
\(\Vert w\Vert _{\infty } \le \ \Vert w\Vert _{H^1(0,1)}, \qquad \forall w \in H^1(0,1),\) 
there holds
\({\Vert v_{n,0}\Vert }_{\infty } \le c_{\ref {cte:v0}} , \qquad \forall n \ge 1.\) 
Setting \(\mu _i=M-z_i c_{\ref {cte:v0}} -|\xi _i^\Gamma |\)  and \(\mu \le \min _{i=1,2} \mu _i\) ,
then \(|\xi _{n,i}-\xi _i^{\Gamma }|\le \mu \)  implies \(|v_{n,i}|\le M\) . Hence \(g_i^{\Gamma ,\mu }(\xi _{n,i}-\xi _i^{\Gamma })\)  can be written as the sum of a linear and a nonlinear functions as follows:
\(g_i^{\Gamma ,\mu }(\xi _{n,i}-\xi _i^{\Gamma })= \left(g_i^{\Gamma ,\mu }\right)^{\prime }(\mu )\, (\xi _{n,i}-\xi _i^{\Gamma })+g_i^{\Gamma ,\mu ,NL}(T_M v_{n,i}+z_i v_{n,0}-\xi _i^{\Gamma }).\) 
By (REF ) we immediately get that
\(\left(g_i^{\Gamma ,\mu }\right)^{\prime }(\mu )\, (\xi _{n,i}-\xi _i^{\Gamma }) \underset{n\rightarrow +\infty }{-\hspace{-4.0pt}-\hspace{-5.0pt}\rightharpoonup } \left(g_i^{\Gamma ,\mu }\right)^{\prime }(\mu )\, (\xi _{i}-\xi _i^{\Gamma }) \hbox{ weakly in }L^2(S).\) 
As for the nonlinear one, from (REF ) and (REF ) we have that \(T_M v_{n,i}+z_i v_{n,0}\)  converges almost everywhere in \(S\times (0,1)\)  to \(T_M v_{i}+z_i v_{0}\) , for \(i=1,2\) . Whence, via the Lebesgue convergence dominated theorem we also obtain that
\(g_i^{\Gamma ,\mu ,NL}(T_M v_{n,i}+z_i v_{n,0}-\xi _i^{\Gamma })\underset{n\rightarrow +\infty }{\longrightarrow }g_i^{\Gamma ,\mu ,NL}(T_M v_{i}+z_i v_{0}-\xi _i^{\Gamma }) \hbox{ strongly in }L^2(S).\) 
Combining (REF ) and (REF ) in (REF ) shows that
\(g_i^{\Gamma ,\mu }(\xi _{n,i}-\xi _i^{\Gamma })) \underset{n\rightarrow +\infty }{-\hspace{-4.0pt}-\hspace{-5.0pt}\rightharpoonup } g_i^{\Gamma ,\mu }(\xi _{i}-\xi _i^{\Gamma })) \hbox{ weakly in }L^2(S).\) 
Therefore, convergences (REF ) and (REF ) lead to
\(\displaystyle \lim _{n\rightarrow +\infty }\sum _{i=1,2}\sum _{\Gamma \in \lbrace 0,1\rbrace }\left[r_i^\Gamma (T_M \tau _n v_{n,i})g_i^{\Gamma ,\mu }(\xi _{n,i}-\xi _i^{\Gamma })\tilde{\xi }_i\right](\Gamma )\\\displaystyle \qquad \qquad =\sum _{i=1,2}\sum _{\Gamma \in \lbrace 0,1\rbrace }\left[r_i^\Gamma (T_M \tau _n v_{i})g_i^{\Gamma ,\mu }(\xi _{i}-\xi _i^{\Gamma })\tilde{\xi }_i\right](\Gamma ).\) 
Then, by collecting (REF ) and (REF ), we obtain the validity of (REF ).
Moreover, since \((u_n,v_n)\)  is a solution to \((P_{M_n})\)  we deduce from (REF ) and (REF ) that
\(\dot{u}+A_M(v,v)= \lim _{n \rightarrow +\infty } \Delta _n u_n + A_M(\tau _n v_n,v)=0.\) 
Eventually, due to (REF ) and (REF ), there holds \(u=E_M v\) , which concludes the proof of the lemma.
